### Determine The Equation of a Line

#### Lesson Objective

This lesson shows you how to determine the equation of a line by using the given information such as the slope, coordinates of a point, etc.

#### About This Lesson

In this lesson, we will learn how to determine the equation of a line using the given information.

This lesson will show you two examples on how to do so using the following information:

- Point (2,5) and Slope = 2
- Points (1,4) and (2,1)

You should proceed by reading the study tips and watch the math video below. After that, you can try out the practice questions.

### Study Tips

#### Tip #1

You need to have some knowledge on the slope-intercept form of a line. You can learn about it by watching the math video in this lesson.

#### Tip #2

For the equation of a line that we want to determine, we would usually want that equation to be in the slope-intercept form (see picture).

This way, we just need to use the given information to find the value of the slope (m) and the y-intercept (b).

After doing so, we can easily determine the equation by substituting m and b with their respective value.

Now, watch the following math video to learn more.

### Math Video

#### Lesson Video

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#### Math Video Transcript

Determine the Equation of a Line Transcript

00:00:01.120

In this lesson, we will learn how to determine the equation of a line, using the available information.

00:00:08.040

Let's look at the first example. Determine the equation of the line, that passes through the points (2, 5), and with the slope of 2.

00:00:19.000

To begin, we should know that the equation of a line can be written in the form of y = mx + b, where m is the slope, and b is the y-intercept.

00:00:30.110

From this, to determine the equation of the line, we just need to find the values of m and b.

00:00:37.120

Now, we can see that, the slope m, is already given as 2.

00:00:43.050

Therefore, we can just substitute m with 2, and the equation now becomes y = 2x + b.

00:00:51.070

Next, we need to find the y-intercept, b.

00:00:55.090

Since the value of b is not given, we need to find it.

00:00:59.220

To do so, we know that the line passes through the point 2, 5.

00:01:06.160

Therefore, we can use this point by substituting x with 2, and substituting y with 5.

00:01:13.240

With this, notice that we can now solve for b.

00:01:18.010

To solve for b, multiply 2 with 2. This gives 4.

00:01:23.100

Next, add negative 4 to both sides of the equation. This gives 5 - 4 = b.

00:01:30.240

5 minus 4 gives 1. Hence, we found the y-intercept, b as 1.

00:01:38.190

With this, we can write b as 1.

00:01:44.230

Finally, since we found both m and b, the equation of the line is y = 2x + 1

00:01:54.060

Now, next example.

00:01:56.240

Determine, the equation of the line, that passes through the points (1,4) and (2,1).

00:02:04.190

Again, we should know that the equation of a line can be written in the form of y = mx + b.

00:02:12.060

We can see that, the slope and y-intercept are not given. Instead, we only have the coordinates of 2 points.

00:02:20.210

With some thinking, we can use these 2 points to find the slope 'm' by applying the slope formula, (y2-y1)/(x2-x1).

00:02:32.140

To use the slope-formula, we can assign this point as point 1, with the x-coordinate as x1, and y-coordinate as y1.

00:02:42.000

Similarly, we assign the next point as point 2, with x-coordinate as x2, and y-coordinate as y2.

00:02:50.230

Now, we can find 'm' by just substituting, y2 with 1, y1 with 4, x2 with 2, and x1 with 1.

00:03:04.100

Alright, we can remove these brackets, as they do nothing.

00:03:09.220

Let's calculate 'm'. Negative multiply by bracket 4 gives negative 4. negative multiply by bracket 1 gives negative 1.

00:03:20.150

1 minus by 4 give negative 3. 2 minus by 1 gives positive 1.

00:03:27.230

Negative 3 divides by positive 1 gives negative 3.

00:03:32.140

So, we found the slope 'm' as negative 3. Now, we can write m as negative 3.

00:03:42.120

Next, we need to find the y-intercept, b.

00:03:48.080

Now, similar to the previous question, we can find b by taking a point on the line, and substitute its x-coordinate and y-coordinate into the equation.

00:03:59.000

Let's take this point 1,4.

00:04:02.200

Substituting x with 1 and y with 4.

00:04:07.220

Now, we can solve for 'b'. Multiplying negative 3 with 1 gives negative 3.

00:04:14.000

Next, add positive 3 to both sides of the equation. This gives 4 + 3 = b.

00:04:21.120

4 plus 3 gives 7. Hence, we find the y-intercept, b is 7.

00:04:29.010

With this, we can now write b as 7.

00:04:34.120

So finally, with both slope and y-intercept found, we have the equation of the line as, y = negative 3x + 7

00:04:44.060

That is all for this lesson. Try out the practice question to further your understanding.

### Practice Questions & More

#### Multiple Choice Questions (MCQ)

Now, let's try some MCQ questions to understand this lesson better.

You can start by going through the series of questions on determining an equation of a line parallel to the x-axis or y-axis or pick your choice of question below.

- Question 1 on determining the equation of a line using the coordinates of two points.

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